Make ToC in the center and left aligned in beamer

A small calculation could advance the margins sufficiently to obtain the centered & left-aligned \tableofcontents display.

\documentclass{beamer}% http://ctan.org/pkg/beamer
\newsavebox{\longestsec}% Box to save longest sectional heading
\begin{document}

\begin{frame}
  \frametitle{Table of contents}
  \begin{lrbox}{\longestsec}Last section in the presentation\end{lrbox}% Capture longest title
  \setlength{\leftskip}{\dimexpr.5\textwidth-.5\wd\longestsec\relax}% Advance left margin accordingly
  \tableofcontents
\end{frame}

\section{First section}
​\begin{frame}
  \frametitle{First slide}
  This is a slide.
\end{frame}

\section{Last section in the presentation}% Longest sectional title
\begin{frame}
  \frametitle{Another slide}
  This is another slide
\end{frame}
\end{document}

​The reason for this round-about way is because beamer collapses the spacing when putting \tableofcontents within a minipage or varwidth (from the varwidth package).

---

Here is an automated way of capturing the width of the longest section:

\documentclass{beamer}% http://ctan.org/pkg/beamer
\usepackage{etoolbox}% http://ctan.org/pkg/etoolbox
\makeatletter
\newlength{\secnamelength}
\newsavebox{\longestsec}% Box to save longest sectional heading
\patchcmd{\beamer@section}% <cmd>
  {\beamer@savemode}% <search>
  {\begin{lrbox}{\longestsec}#1\end{lrbox}%
   \ifdim\wd\longestsec>\secnamelength\relax\setlength{\secnamelength}{\wd\longestsec}\fi%
   \beamer@savemode}% <replace>
  {}{}% <success><failure>
\AtEndDocument{% http://tex.stackexchange.com/q/137495/5764
  \immediate\write\@auxout{\global\secnamelength=\the\secnamelength}%
}
\makeatother
\begin{document}

\begin{frame}
  \frametitle{Table of contents}
  \setlength{\leftskip}{\dimexpr.5\textwidth-.5\secnamelength\relax}% Advance left margin accordingly
  \tableofcontents
\end{frame}

\section{First section}
​\begin{frame}
  \frametitle{First slide}
  This is a slide.
\end{frame}

\section{Last section in the presentation}% Longest sectional title
\begin{frame}
  \frametitle{Another slide}
  This is another slide
\end{frame}
\end{document}

The above stores the length of the longest section in \secnamelength that is written to the .aux at the end of the document. A patch of \beamer@section (with the aid of etoolbox is necessary to grab the maximum length of a section name and drop it into the .aux \AtEndDocument so it can be retrieved in a successive run.

By patching \beamer@sectionintoc command, I could use varwidth inside center environment to make ToC lines left aligned and in the center.

The only limitation of this solution is, ToC lines could not be wrapped.

\documentclass{beamer}    
\usepackage{varwidth}    
\usepackage{etoolbox}
\makeatletter
%\patchcmd{\beamer@sectionintoc}{\vskip1.5em}{\vskip0.5em}{}{}
\patchcmd{\beamer@sectionintoc}{%
  \hbox{\vbox{%
    \def\beamer@breakhere{\\}%
    \beamer@tocact{\ifnum\c@section=#1\beamer@toc@cs\else\beamer@toc@os\fi}{section in toc}}}%
}{%
  \hbox{%
    \def\beamer@breakhere{}%
    \beamer@tocact{\ifnum\c@section=#1\beamer@toc@cs\else\beamer@toc@os\fi}{section in toc}}%
}{}{}
\makeatother    
\setbeamertemplate{section in toc}[sections numbered]    
\begin{document}

\begin{frame}
\frametitle{Table of contents}
\begin{center}
\begin{varwidth}{\textwidth}
\tableofcontents[sectionstyle=show,subsectionstyle=hide] 
\end{varwidth}
\end{center}
\end{frame}

\section{First section}    
​\begin{frame}
\frametitle{First slide}
This is a slide.
\end{frame}

\section{Second section in the presentation}    
\begin{frame}
\frametitle{Another slide}
This is another slide.
\end{frame}

\section{Last section}    
\begin{frame}
\frametitle{Last slide}
This is the last slide.
\end{frame}

\end{document}

As shown in my comment, the automatism provided by Werner can be problematic if you have short names like \section[shortName]{longName} e.g. for beamer slide headlines. It always takes the first argument, which would then be the short name.

I implemented a quick solution now (with etoolbox and ifthen), which will take the second argument if provided:

\makeatletter
\newlength{\secnamelength}
\newsavebox{\longestsec}% Box to save longest sectional heading
\patchcmd{\beamer@section}% <cmd>
{\beamer@savemode}% <search>
{
\ifstrempty{#2}{%
    \begin{lrbox}{\longestsec}#1\end{lrbox}
    \ifthenelse{
        \wd\longestsec > \secnamelength
    }{
        \setlength{\secnamelength}{\wd\longestsec}
    }{}%else
}{
    \begin{lrbox}{\longestsec}#2\end{lrbox}
    \ifthenelse{
        \wd\longestsec > \secnamelength
    }{
        \setlength{\secnamelength}{\wd\longestsec}
    }{}%else
}
\beamer@savemode
}% <replace>
{}{}% <success><failure>
\AtEndDocument{% http://tex.stackexchange.com/q/137495/5764
    \immediate\write\@auxout{\global\secnamelength=\the\secnamelength}%
}
\makeatother